Single-ownership ("affine types") is a separate concept from a borrow checker. Your language doesn't need a borrow checker (or references at all) to benefit from single-ownership, though it may make some patterns more convenient or efficient.
If we consider a Rust-like language without the borrow checker, it's obviously still Turing-complete. For functions that take references as parameters, instead you would simply pass ownership of the value back to the caller as part of the return value. And for structs that hold references, you would instead have them hold reference-counted handles. The former case is merely less convenient, and the latter case is merely less efficient.